P. 148 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14701-14800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	hmeoopn 14701	Homeomorphisms preserve openness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
|- X = U. J   =>    |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( A e. J <-> ( F " A ) e. K ) )
Theorem	hmeocld 14702	Homeomorphisms preserve closedness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
|- X = U. J   =>    |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( A e. ( Clsd ` J ) <-> ( F " A ) e. ( Clsd ` K ) ) )
Theorem	hmeontr 14703	Homeomorphisms preserve interiors. (Contributed by Mario Carneiro, 25-Aug-2015.)
|- X = U. J   =>    |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( int ` K ) ` ( F " A ) ) = ( F " ( ( int ` J ) ` A ) ) )
Theorem	hmeoimaf1o 14704*	The function mapping open sets to their images under a homeomorphism is a bijection of topologies. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- G = ( x e. J |-> ( F " x ) )   =>    |- ( F e. ( J Homeo K ) -> G : J -1-1-onto-> K )
Theorem	hmeores 14705	The restriction of a homeomorphism is a homeomorphism. (Contributed by Mario Carneiro, 14-Sep-2014.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
|- X = U. J   =>    |- ( ( F e. ( J Homeo K ) /\ Y C_ X ) -> ( F |` Y ) e. ( ( J ↾t Y ) Homeo ( K ↾t ( F " Y ) ) ) )
Theorem	hmeoco 14706	The composite of two homeomorphisms is a homeomorphism. (Contributed by FL, 9-Mar-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
|- ( ( F e. ( J Homeo K ) /\ G e. ( K Homeo L ) ) -> ( G o. F ) e. ( J Homeo L ) )
Theorem	idhmeo 14707	The identity function is a homeomorphism. (Contributed by FL, 14-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
|- ( J e. (TopOn ` X ) -> ( _I |` X ) e. ( J Homeo J ) )
Theorem	hmeocnvb 14708	The converse of a homeomorphism is a homeomorphism. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
|- ( Rel F -> ( `' F e. ( J Homeo K ) <-> F e. ( K Homeo J ) ) )
Theorem	txhmeo 14709*	Lift a pair of homeomorphisms on the factors to a homeomorphism of product topologies. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- X = U. J   &    |- Y = U. K   &    |- ( ph -> F e. ( J Homeo L ) )   &    |- ( ph -> G e. ( K Homeo M ) )   =>    |- ( ph -> ( x e. X , y e. Y |-> <. ( F ` x ) , ( G ` y ) >. ) e. ( ( J tX K ) Homeo ( L tX M ) ) )
Theorem	txswaphmeolem 14710*	Show inverse for the "swap components" operation on a Cartesian product. (Contributed by Mario Carneiro, 21-Mar-2015.)
|- ( ( y e. Y , x e. X |-> <. x , y >. ) o. ( x e. X , y e. Y |-> <. y , x >. ) ) = ( _I |` ( X X. Y ) )
Theorem	txswaphmeo 14711*	There is a homeomorphism from X X. Y to Y X. X. (Contributed by Mario Carneiro, 21-Mar-2015.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( x e. X , y e. Y |-> <. y , x >. ) e. ( ( J tX K ) Homeo ( K tX J ) ) )
9.2  Metric spaces
9.2.1  Pseudometric spaces
Theorem	psmetrel 14712	The class of pseudometrics is a relation. (Contributed by Jim Kingdon, 24-Apr-2023.)
|- Rel PsMet
Theorem	ispsmet 14713*	Express the predicate " D is a pseudometric". (Contributed by Thierry Arnoux, 7-Feb-2018.)
|- ( X e. V -> ( D e. (PsMet ` X ) <-> ( D : ( X X. X ) --> RR* /\ A. x e. X ( ( x D x ) = 0 /\ A. y e. X A. z e. X ( x D y ) <_ ( ( z D x ) +e ( z D y ) ) ) ) ) )
Theorem	psmetdmdm 14714	Recover the base set from a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
|- ( D e. (PsMet ` X ) -> X = dom dom D )
Theorem	psmetf 14715	The distance function of a pseudometric as a function. (Contributed by Thierry Arnoux, 7-Feb-2018.)
|- ( D e. (PsMet ` X ) -> D : ( X X. X ) --> RR* )
Theorem	psmetcl 14716	Closure of the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 7-Feb-2018.)
|- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) e. RR* )
Theorem	psmet0 14717	The distance function of a pseudometric space is zero if its arguments are equal. (Contributed by Thierry Arnoux, 7-Feb-2018.)
|- ( ( D e. (PsMet ` X ) /\ A e. X ) -> ( A D A ) = 0 )
Theorem	psmettri2 14718	Triangle inequality for the distance function of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.)
|- ( ( D e. (PsMet ` X ) /\ ( C e. X /\ A e. X /\ B e. X ) ) -> ( A D B ) <_ ( ( C D A ) +e ( C D B ) ) )
Theorem	psmetsym 14719	The distance function of a pseudometric is symmetrical. (Contributed by Thierry Arnoux, 7-Feb-2018.)
|- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) = ( B D A ) )
Theorem	psmettri 14720	Triangle inequality for the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 11-Feb-2018.)
|- ( ( D e. (PsMet ` X ) /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A D B ) <_ ( ( A D C ) +e ( C D B ) ) )
Theorem	psmetge0 14721	The distance function of a pseudometric space is nonnegative. (Contributed by Thierry Arnoux, 7-Feb-2018.) (Revised by Jim Kingdon, 19-Apr-2023.)
|- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> 0 <_ ( A D B ) )
Theorem	psmetxrge0 14722	The distance function of a pseudometric space is a function into the nonnegative extended real numbers. (Contributed by Thierry Arnoux, 24-Feb-2018.)
|- ( D e. (PsMet ` X ) -> D : ( X X. X ) --> ( 0 [,] +oo ) )
Theorem	psmetres2 14723	Restriction of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.)
|- ( ( D e. (PsMet ` X ) /\ R C_ X ) -> ( D |` ( R X. R ) ) e. (PsMet ` R ) )
Theorem	psmetlecl 14724	Real closure of an extended metric value that is upper bounded by a real. (Contributed by Thierry Arnoux, 11-Mar-2018.)
|- ( ( D e. (PsMet ` X ) /\ ( A e. X /\ B e. X ) /\ ( C e. RR /\ ( A D B ) <_ C ) ) -> ( A D B ) e. RR )
Theorem	distspace 14725	A set X together with a (distance) function D which is a pseudometric is a distance space (according to E. Deza, M.M. Deza: "Dictionary of Distances", Elsevier, 2006), i.e. a (base) set X equipped with a distance D, which is a mapping of two elements of the base set to the (extended) reals and which is nonnegative, symmetric and equal to 0 if the two elements are equal. (Contributed by AV, 15-Oct-2021.) (Revised by AV, 5-Jul-2022.)
|- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( ( D : ( X X. X ) --> RR* /\ ( A D A ) = 0 ) /\ ( 0 <_ ( A D B ) /\ ( A D B ) = ( B D A ) ) ) )
9.2.2  Basic metric space properties
Syntax	cxms 14726	Extend class notation with the class of extended metric spaces.
class *MetSp
Syntax	cms 14727	Extend class notation with the class of metric spaces.
class MetSp
Syntax	ctms 14728	Extend class notation with the function mapping a metric to the metric space it defines.
class toMetSp
Definition	df-xms 14729	Define the (proper) class of extended metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- *MetSp = { f e. TopSp | ( TopOpen ` f ) = ( MetOpen ` ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) ) }
Definition	df-ms 14730	Define the (proper) class of metric spaces. (Contributed by NM, 27-Aug-2006.)
|- MetSp = { f e. *MetSp | ( ( dist ` f ) |` ( ( Base ` f ) X. ( Base ` f ) ) ) e. ( Met ` ( Base ` f ) ) }
Definition	df-tms 14731	Define the function mapping a metric to the metric space which it defines. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- toMetSp = ( d e. U. ran *Met |-> ( { <. ( Base ` ndx ) , dom dom d >. , <. ( dist ` ndx ) , d >. } sSet <. (TopSet ` ndx ) , ( MetOpen ` d ) >. ) )
Theorem	metrel 14732	The class of metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.)
|- Rel Met
Theorem	xmetrel 14733	The class of extended metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.)
|- Rel *Met
Theorem	ismet 14734*	Express the predicate " D is a metric". (Contributed by NM, 25-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
|- ( X e. A -> ( D e. ( Met ` X ) <-> ( D : ( X X. X ) --> RR /\ A. x e. X A. y e. X ( ( ( x D y ) = 0 <-> x = y ) /\ A. z e. X ( x D y ) <_ ( ( z D x ) + ( z D y ) ) ) ) ) )
Theorem	isxmet 14735*	Express the predicate " D is an extended metric". (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( X e. A -> ( D e. ( *Met ` X ) <-> ( D : ( X X. X ) --> RR* /\ A. x e. X A. y e. X ( ( ( x D y ) = 0 <-> x = y ) /\ A. z e. X ( x D y ) <_ ( ( z D x ) +e ( z D y ) ) ) ) ) )
Theorem	ismeti 14736*	Properties that determine a metric. (Contributed by NM, 17-Nov-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
|- X e. _V   &    |- D : ( X X. X ) --> RR   &    |- ( ( x e. X /\ y e. X ) -> ( ( x D y ) = 0 <-> x = y ) )   &    |- ( ( x e. X /\ y e. X /\ z e. X ) -> ( x D y ) <_ ( ( z D x ) + ( z D y ) ) )   =>    |- D e. ( Met ` X )
Theorem	isxmetd 14737*	Properties that determine an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ph -> X e. _V )   &    |- ( ph -> D : ( X X. X ) --> RR* )   &    |- ( ( ph /\ ( x e. X /\ y e. X ) ) -> ( ( x D y ) = 0 <-> x = y ) )   &    |- ( ( ph /\ ( x e. X /\ y e. X /\ z e. X ) ) -> ( x D y ) <_ ( ( z D x ) +e ( z D y ) ) )   =>    |- ( ph -> D e. ( *Met ` X ) )
Theorem	isxmet2d 14738*	It is safe to only require the triangle inequality when the values are real (so that we can use the standard addition over the reals), but in this case the nonnegativity constraint cannot be deduced and must be provided separately. (Counterexample: D ( x , y ) = if ( x = y , 0 , -oo ) satisfies all hypotheses except nonnegativity.) (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ph -> X e. _V )   &    |- ( ph -> D : ( X X. X ) --> RR* )   &    |- ( ( ph /\ ( x e. X /\ y e. X ) ) -> 0 <_ ( x D y ) )   &    |- ( ( ph /\ ( x e. X /\ y e. X ) ) -> ( ( x D y ) <_ 0 <-> x = y ) )   &    |- ( ( ph /\ ( x e. X /\ y e. X /\ z e. X ) /\ ( ( z D x ) e. RR /\ ( z D y ) e. RR ) ) -> ( x D y ) <_ ( ( z D x ) + ( z D y ) ) )   =>    |- ( ph -> D e. ( *Met ` X ) )
Theorem	metflem 14739*	Lemma for metf 14741 and others. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
|- ( D e. ( Met ` X ) -> ( D : ( X X. X ) --> RR /\ A. x e. X A. y e. X ( ( ( x D y ) = 0 <-> x = y ) /\ A. z e. X ( x D y ) <_ ( ( z D x ) + ( z D y ) ) ) ) )
Theorem	xmetf 14740	Mapping of the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( D e. ( *Met ` X ) -> D : ( X X. X ) --> RR* )
Theorem	metf 14741	Mapping of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.)
|- ( D e. ( Met ` X ) -> D : ( X X. X ) --> RR )
Theorem	xmetcl 14742	Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.)
|- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) e. RR* )
Theorem	metcl 14743	Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.)
|- ( ( D e. ( Met ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) e. RR )
Theorem	ismet2 14744	An extended metric is a metric exactly when it takes real values for all values of the arguments. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( D e. ( Met ` X ) <-> ( D e. ( *Met ` X ) /\ D : ( X X. X ) --> RR ) )
Theorem	metxmet 14745	A metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( D e. ( Met ` X ) -> D e. ( *Met ` X ) )
Theorem	xmetdmdm 14746	Recover the base set from an extended metric. (Contributed by Mario Carneiro, 23-Aug-2015.)
|- ( D e. ( *Met ` X ) -> X = dom dom D )
Theorem	metdmdm 14747	Recover the base set from a metric. (Contributed by Mario Carneiro, 23-Aug-2015.)
|- ( D e. ( Met ` X ) -> X = dom dom D )
Theorem	xmetunirn 14748	Two ways to express an extended metric on an unspecified base. (Contributed by Mario Carneiro, 13-Oct-2015.)
|- ( D e. U. ran *Met <-> D e. ( *Met ` dom dom D ) )
Theorem	xmeteq0 14749	The value of an extended metric is zero iff its arguments are equal. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( ( A D B ) = 0 <-> A = B ) )
Theorem	meteq0 14750	The value of a metric is zero iff its arguments are equal. Property M2 of [Kreyszig] p. 4. (Contributed by NM, 30-Aug-2006.)
|- ( ( D e. ( Met ` X ) /\ A e. X /\ B e. X ) -> ( ( A D B ) = 0 <-> A = B ) )
Theorem	xmettri2 14751	Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( D e. ( *Met ` X ) /\ ( C e. X /\ A e. X /\ B e. X ) ) -> ( A D B ) <_ ( ( C D A ) +e ( C D B ) ) )
Theorem	mettri2 14752	Triangle inequality for the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 20-Aug-2015.)
|- ( ( D e. ( Met ` X ) /\ ( C e. X /\ A e. X /\ B e. X ) ) -> ( A D B ) <_ ( ( C D A ) + ( C D B ) ) )
Theorem	xmet0 14753	The distance function of a metric space is zero if its arguments are equal. Definition 14-1.1(a) of [Gleason] p. 223. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( D e. ( *Met ` X ) /\ A e. X ) -> ( A D A ) = 0 )
Theorem	met0 14754	The distance function of a metric space is zero if its arguments are equal. Definition 14-1.1(a) of [Gleason] p. 223. (Contributed by NM, 30-Aug-2006.)
|- ( ( D e. ( Met ` X ) /\ A e. X ) -> ( A D A ) = 0 )
Theorem	xmetge0 14755	The distance function of a metric space is nonnegative. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> 0 <_ ( A D B ) )
Theorem	metge0 14756	The distance function of a metric space is nonnegative. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
|- ( ( D e. ( Met ` X ) /\ A e. X /\ B e. X ) -> 0 <_ ( A D B ) )
Theorem	xmetlecl 14757	Real closure of an extended metric value that is upper bounded by a real. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( D e. ( *Met ` X ) /\ ( A e. X /\ B e. X ) /\ ( C e. RR /\ ( A D B ) <_ C ) ) -> ( A D B ) e. RR )
Theorem	xmetsym 14758	The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) = ( B D A ) )
Theorem	xmetpsmet 14759	An extended metric is a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
|- ( D e. ( *Met ` X ) -> D e. (PsMet ` X ) )
Theorem	xmettpos 14760	The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( D e. ( *Met ` X ) -> tpos D = D )
Theorem	metsym 14761	The distance function of a metric space is symmetric. Definition 14-1.1(c) of [Gleason] p. 223. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 20-Aug-2015.)
|- ( ( D e. ( Met ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) = ( B D A ) )
Theorem	xmettri 14762	Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( D e. ( *Met ` X ) /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A D B ) <_ ( ( A D C ) +e ( C D B ) ) )
Theorem	mettri 14763	Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by NM, 27-Aug-2006.)
|- ( ( D e. ( Met ` X ) /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A D B ) <_ ( ( A D C ) + ( C D B ) ) )
Theorem	xmettri3 14764	Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( D e. ( *Met ` X ) /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A D B ) <_ ( ( A D C ) +e ( B D C ) ) )
Theorem	mettri3 14765	Triangle inequality for the distance function of a metric space. (Contributed by NM, 13-Mar-2007.)
|- ( ( D e. ( Met ` X ) /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A D B ) <_ ( ( A D C ) + ( B D C ) ) )
Theorem	xmetrtri 14766	One half of the reverse triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 4-Sep-2015.)
|- ( ( D e. ( *Met ` X ) /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A D C ) +e -e ( B D C ) ) <_ ( A D B ) )
Theorem	metrtri 14767	Reverse triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Jim Kingdon, 21-Apr-2023.)
|- ( ( D e. ( Met ` X ) /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( abs ` ( ( A D C ) - ( B D C ) ) ) <_ ( A D B ) )
Theorem	metn0 14768	A metric space is nonempty iff its base set is nonempty. (Contributed by NM, 4-Oct-2007.) (Revised by Mario Carneiro, 14-Aug-2015.)
|- ( D e. ( Met ` X ) -> ( D =/= (/) <-> X =/= (/) ) )
Theorem	xmetres2 14769	Restriction of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- ( ( D e. ( *Met ` X ) /\ R C_ X ) -> ( D |` ( R X. R ) ) e. ( *Met ` R ) )
Theorem	metreslem 14770	Lemma for metres 14773. (Contributed by Mario Carneiro, 24-Aug-2015.)
|- ( dom D = ( X X. X ) -> ( D |` ( R X. R ) ) = ( D |` ( ( X i^i R ) X. ( X i^i R ) ) ) )
Theorem	metres2 14771	Lemma for metres 14773. (Contributed by FL, 12-Oct-2006.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)
|- ( ( D e. ( Met ` X ) /\ R C_ X ) -> ( D |` ( R X. R ) ) e. ( Met ` R ) )
Theorem	xmetres 14772	A restriction of an extended metric is an extended metric. (Contributed by Mario Carneiro, 24-Aug-2015.)
|- ( D e. ( *Met ` X ) -> ( D |` ( R X. R ) ) e. ( *Met ` ( X i^i R ) ) )
Theorem	metres 14773	A restriction of a metric is a metric. (Contributed by NM, 26-Aug-2007.) (Revised by Mario Carneiro, 14-Aug-2015.)
|- ( D e. ( Met ` X ) -> ( D |` ( R X. R ) ) e. ( Met ` ( X i^i R ) ) )
Theorem	0met 14774	The empty metric. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
|- (/) e. ( Met ` (/) )
9.2.3  Metric space balls
Theorem	blfvalps 14775*	The value of the ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Feb-2018.)
|- ( D e. (PsMet ` X ) -> ( ball ` D ) = ( x e. X , r e. RR* |-> { y e. X | ( x D y ) < r } ) )
Theorem	blfval 14776*	The value of the ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Proof shortened by Thierry Arnoux, 11-Feb-2018.)
|- ( D e. ( *Met ` X ) -> ( ball ` D ) = ( x e. X , r e. RR* |-> { y e. X | ( x D y ) < r } ) )
Theorem	blex 14777	A ball is a set. Also see blfn 14231 in case you just know D is a set, not D e. ( *Met ` X ). (Contributed by Jim Kingdon, 4-May-2023.)
|- ( D e. ( *Met ` X ) -> ( ball ` D ) e. _V )
Theorem	blvalps 14778*	The ball around a point P is the set of all points whose distance from P is less than the ball's radius R. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
|- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) = { x e. X | ( P D x ) < R } )
Theorem	blval 14779*	The ball around a point P is the set of all points whose distance from P is less than the ball's radius R. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
|- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) = { x e. X | ( P D x ) < R } )
Theorem	elblps 14780	Membership in a ball. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
|- ( ( D e. (PsMet ` X ) /\ P e. X /\ R e. RR* ) -> ( A e. ( P ( ball ` D ) R ) <-> ( A e. X /\ ( P D A ) < R ) ) )
Theorem	elbl 14781	Membership in a ball. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
|- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( A e. ( P ( ball ` D ) R ) <-> ( A e. X /\ ( P D A ) < R ) ) )
Theorem	elbl2ps 14782	Membership in a ball. (Contributed by NM, 9-Mar-2007.) (Revised by Thierry Arnoux, 11-Mar-2018.)
|- ( ( ( D e. (PsMet ` X ) /\ R e. RR* ) /\ ( P e. X /\ A e. X ) ) -> ( A e. ( P ( ball ` D ) R ) <-> ( P D A ) < R ) )
Theorem	elbl2 14783	Membership in a ball. (Contributed by NM, 9-Mar-2007.)
|- ( ( ( D e. ( *Met ` X ) /\ R e. RR* ) /\ ( P e. X /\ A e. X ) ) -> ( A e. ( P ( ball ` D ) R ) <-> ( P D A ) < R ) )
Theorem	elbl3ps 14784	Membership in a ball, with reversed distance function arguments. (Contributed by NM, 10-Nov-2007.)
|- ( ( ( D e. (PsMet ` X ) /\ R e. RR* ) /\ ( P e. X /\ A e. X ) ) -> ( A e. ( P ( ball ` D ) R ) <-> ( A D P ) < R ) )
Theorem	elbl3 14785	Membership in a ball, with reversed distance function arguments. (Contributed by NM, 10-Nov-2007.)
|- ( ( ( D e. ( *Met ` X ) /\ R e. RR* ) /\ ( P e. X /\ A e. X ) ) -> ( A e. ( P ( ball ` D ) R ) <-> ( A D P ) < R ) )
Theorem	blcomps 14786	Commute the arguments to the ball function. (Contributed by Mario Carneiro, 22-Jan-2014.) (Revised by Thierry Arnoux, 11-Mar-2018.)
|- ( ( ( D e. (PsMet ` X ) /\ R e. RR* ) /\ ( P e. X /\ A e. X ) ) -> ( A e. ( P ( ball ` D ) R ) <-> P e. ( A ( ball ` D ) R ) ) )
Theorem	blcom 14787	Commute the arguments to the ball function. (Contributed by Mario Carneiro, 22-Jan-2014.)
|- ( ( ( D e. ( *Met ` X ) /\ R e. RR* ) /\ ( P e. X /\ A e. X ) ) -> ( A e. ( P ( ball ` D ) R ) <-> P e. ( A ( ball ` D ) R ) ) )
Theorem	xblpnfps 14788	The infinity ball in an extended metric is the set of all points that are a finite distance from the center. (Contributed by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
|- ( ( D e. (PsMet ` X ) /\ P e. X ) -> ( A e. ( P ( ball ` D ) +oo ) <-> ( A e. X /\ ( P D A ) e. RR ) ) )
Theorem	xblpnf 14789	The infinity ball in an extended metric is the set of all points that are a finite distance from the center. (Contributed by Mario Carneiro, 23-Aug-2015.)
|- ( ( D e. ( *Met ` X ) /\ P e. X ) -> ( A e. ( P ( ball ` D ) +oo ) <-> ( A e. X /\ ( P D A ) e. RR ) ) )
Theorem	blpnf 14790	The infinity ball in a standard metric is just the whole space. (Contributed by Mario Carneiro, 23-Aug-2015.)
|- ( ( D e. ( Met ` X ) /\ P e. X ) -> ( P ( ball ` D ) +oo ) = X )
Theorem	bldisj 14791	Two balls are disjoint if the center-to-center distance is more than the sum of the radii. (Contributed by Mario Carneiro, 30-Dec-2013.)
|- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ Q e. X ) /\ ( R e. RR* /\ S e. RR* /\ ( R +e S ) <_ ( P D Q ) ) ) -> ( ( P ( ball ` D ) R ) i^i ( Q ( ball ` D ) S ) ) = (/) )
Theorem	blgt0 14792	A nonempty ball implies that the radius is positive. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
|- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) /\ A e. ( P ( ball ` D ) R ) ) -> 0 < R )
Theorem	bl2in 14793	Two balls are disjoint if they don't overlap. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
|- ( ( ( D e. ( Met ` X ) /\ P e. X /\ Q e. X ) /\ ( R e. RR /\ R <_ ( ( P D Q ) / 2 ) ) ) -> ( ( P ( ball ` D ) R ) i^i ( Q ( ball ` D ) R ) ) = (/) )
Theorem	xblss2ps 14794	One ball is contained in another if the center-to-center distance is less than the difference of the radii. In this version of blss2 14797 for extended metrics, we have to assume the balls are a finite distance apart, or else P will not even be in the infinity ball around Q. (Contributed by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
|- ( ph -> D e. (PsMet ` X ) )   &    |- ( ph -> P e. X )   &    |- ( ph -> Q e. X )   &    |- ( ph -> R e. RR* )   &    |- ( ph -> S e. RR* )   &    |- ( ph -> ( P D Q ) e. RR )   &    |- ( ph -> ( P D Q ) <_ ( S +e -e R ) )   =>    |- ( ph -> ( P ( ball ` D ) R ) C_ ( Q ( ball ` D ) S ) )
Theorem	xblss2 14795	One ball is contained in another if the center-to-center distance is less than the difference of the radii. In this version of blss2 14797 for extended metrics, we have to assume the balls are a finite distance apart, or else P will not even be in the infinity ball around Q. (Contributed by Mario Carneiro, 23-Aug-2015.)
|- ( ph -> D e. ( *Met ` X ) )   &    |- ( ph -> P e. X )   &    |- ( ph -> Q e. X )   &    |- ( ph -> R e. RR* )   &    |- ( ph -> S e. RR* )   &    |- ( ph -> ( P D Q ) e. RR )   &    |- ( ph -> ( P D Q ) <_ ( S +e -e R ) )   =>    |- ( ph -> ( P ( ball ` D ) R ) C_ ( Q ( ball ` D ) S ) )
Theorem	blss2ps 14796	One ball is contained in another if the center-to-center distance is less than the difference of the radii. (Contributed by Mario Carneiro, 15-Jan-2014.) (Revised by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
|- ( ( ( D e. (PsMet ` X ) /\ P e. X /\ Q e. X ) /\ ( R e. RR /\ S e. RR /\ ( P D Q ) <_ ( S - R ) ) ) -> ( P ( ball ` D ) R ) C_ ( Q ( ball ` D ) S ) )
Theorem	blss2 14797	One ball is contained in another if the center-to-center distance is less than the difference of the radii. (Contributed by Mario Carneiro, 15-Jan-2014.) (Revised by Mario Carneiro, 23-Aug-2015.)
|- ( ( ( D e. ( *Met ` X ) /\ P e. X /\ Q e. X ) /\ ( R e. RR /\ S e. RR /\ ( P D Q ) <_ ( S - R ) ) ) -> ( P ( ball ` D ) R ) C_ ( Q ( ball ` D ) S ) )
Theorem	blhalf 14798	A ball of radius R / 2 is contained in a ball of radius R centered at any point inside the smaller ball. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jan-2014.)
|- ( ( ( M e. ( *Met ` X ) /\ Y e. X ) /\ ( R e. RR /\ Z e. ( Y ( ball ` M ) ( R / 2 ) ) ) ) -> ( Y ( ball ` M ) ( R / 2 ) ) C_ ( Z ( ball ` M ) R ) )
Theorem	blfps 14799	Mapping of a ball. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
|- ( D e. (PsMet ` X ) -> ( ball ` D ) : ( X X. RR* ) --> ~P X )
Theorem	blf 14800	Mapping of a ball. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
|- ( D e. ( *Met ` X ) -> ( ball ` D ) : ( X X. RR* ) --> ~P X )